<<

. 6
( 39)



>>


+ Prob CFt=1 will be case 2 · CFt=1 cash ¬‚ow in case 2

+ Prob CFt=1 will be case 3 · CFt=1 cash ¬‚ow in case 3 .

This expected return of $206.80 is less than the $210 that the government promises. Put
di¬erently, if I promise you a rate of return of 5%,
$210 ’ $200
Promised(˜t=0,1 ) = = 5.00%
r
$200
(5.11)
˜
Promised(Ct=1 ) ’ CFt=0
Promised(˜t=0,1 ) =
r ,
CFt=0

your expected rate of return is only
$206.80 ’ $200
E (˜t=0,t=1 ) = = 3.40%
r
$200
(5.12)
˜
E (Ct=1 ) ’ CFt=0
E (˜t=0,t=1 ) =
r ,
CFt=0

which is less than the 5% interest rate that Uncle Sam promises”and surely delivers.
You need to determine how much I have to promise you to “break even,” so that you expect to Determine how much
more interest promise
end up with the same $210 that you could receive from Uncle Sam. In Table 5.1, we computed
you need to break even.
the expected payo¬ as the probability-weighted average payo¬. You want this payo¬ to be not
¬le=uncertainty.tex: LP
90 Chapter 5. Uncertainty, Default, and Risk.

$206.80, but the $210 that you can get if you put your money in government bonds. So, you
now solve for an amount x that you want to receive if I have money,

˜
E (Ct=1 ) = · x
98%

+ ·
1% $100

+ · = $210.00
1% $0
(5.13)
˜
E (Ct=1 ) = Prob CFt=1 will be case 1 · CFt=1 cash ¬‚ow in case 1

+ Prob CFt=1 will be case 2 · CFt=1 cash ¬‚ow in case 2

+ Prob CFt=1 will be case 3 · CFt=1 cash ¬‚ow in case 3 .

The solution is that if I promise you x = $213.27, you will expect to earn the same 5% interest
rate that you can earn in Treasury bonds. Table 5.2 con¬rms that a promise of $213.27 for a
cash investment of $200, which is a promised interest rate of
$213.27 ’ $200
Promised(˜t=0,1 ) = = 6.63%
r
$200
(5.14)
˜
Promised(Ct=1 ) ’ CFt=0
Promised(˜t=0,1 ) =
r ,
CFt=0

and provides an expected interest rate of

(5.15)
E (˜t=0,1 ) = 98% · (+6.63%) + 1% · (’50%) + 1% · (’100%) = 5% .
r




Table 5.2. Risky Payo¬ Table: 6.63% Promised Interest Rate

How Likely
Receive Cash which is a
Flow CFt=1 return of (Probability)
$213.27 +6.63% 98% of the time
$100.00 “50.00% 1% of the time
$0.00 “100.00% 1% of the time
$210.00 +5.00% in expectation




The di¬erence of 1.63% between the promised (or quoted) interest rate of 6.63% and the ex-
The difference between
the promised and pected interest rate of 5% is the default premium”it is the extra interest rate that is caused by
expected interest rate is
the default risk. Of course, you only receive this 6.63% if everything goes perfectly. In our
the default premium.
perfect world with risk-neutral investors,

= +
6.63% 5% 1.63%
(5.16)
“Promised Interest Rate” = “Time Premium” + “Default Premium” .




Important: Except for 100% safe bonds (Treasuries), the promised (or quoted)
rate of return is higher than the expected rate of return. Never confuse the higher
promised rate for the lower expected rate.
Financial securities and information providers rarely, if ever, provide expected
rates of return.
¬le=uncertainty.tex: RP
91
Section 5·2. Interest Rates and Credit Risk (Default Risk).

On average, the expected rate of return is the expected time premium plus the expected default In a risk-neutral world,
all securities have the
premium. Because the expected default premium is zero on average,
same exp. rate of return.

E Rate of Return = E Time Premium + E Realized Default Premium
(5.17)
= E Time Premium + .
0

If you want to work this out, you can compute the expected realized default premium as follows:
you will receive (6.63% ’ 5% = 1.63%) in 98% of all cases; ’50% ’ 5% = ’55% in 1% of all cases
(note that you lose the time-premium); and ’100% ’ 5% = ’105% in the remaining 1% of all
cases (i.e., you lose not only all your money, but also the time-premium). Therefore,

(5.18)
E Realized Default Premium = 98% · (+1.63%) + 1% · (’55%) + 1% · (’105%) = 0% .



Solve Now!
Q 5.6 Recompute the example in Table 5.2 assuming that the probability of receiving full pay-
ment of $210 is only 95%, the probability of receiving $100 is 1%, and the probability of receiving
absolutely no payment is 4%.

(a) At the promised interest rate of 5%, what is the expected interest rate?

(b) What interest rate is required as a promise to ensure an expected interest rate of 5%?



5·2.C. Preview: Risk-Averse Investors Have Demanded Higher Expected Rates

We have assumed that investors are risk-neutral”indi¬erent between two loans that have the In addition to the
default premium, in real
same expected rate of return. As we have already mentioned, in the real world, risk-averse
life, investors also
investors would demand and expect to receive a little bit more for the risky loan. Would you demand a risk premium.
rather invest into a bond that is known to pay o¬ 5% (for example, a U.S. government bond), or
would you rather invest in a bond that is “merely” expected to pay o¬ 5% (such as my 6.63%
bond)? Like most lenders, you are likely to be better o¬ if you know exactly how much you will
receive, rather than live with the uncertainty of my situation. Thus, as a risk-averse investor,
you would probably ask me not only for the higher promised interest rate of 6.63%, which only
gets you to an expected interest rate of 5%, but an even higher promise in order to get you more
than 6.63%. For example, you might demand 6.75%, in which case you would expect to earn
not just 5%, but a little more. The extra 12 basis points is called a risk premium, and it is an
interest component required above and beyond the time premium (i.e., what the U.S. Treasury
Department pays for use of money over time) and above and beyond the default premium (i.e.,
what the promised interest has to be for you to just expect to receive the same rate of return
as what the government o¬ers).
Recapping, we know that 5% is the time-value of money that you can earn in interest from the A more general
decomposition of rates
Treasury. You also know that 1.63% is the extra default premium that I must promise you, a
of return.
risk-neutral lender, to allow you to expect to earn 5%, given that repayment is not guaranteed.
Finally, if you are not risk-neutral but risk-averse, I may have to pay even more than 6.63%,
although we do not know exactly how much.
If you want, you could think of further interest decompositions. It could even be that the time- More intellectually
interesting, but
premium is itself determined by other factors (such as your preference between consuming
otherwise not too useful
today and consuming next year, the in¬‚ation rate, taxes, or other issues, that we are brushing decompositions.
over). Then there would be a liquidity premium, an extra interest rate that a lender would
demand if the bond could not easily be sold”resale is much easier with Treasury bonds.
¬le=uncertainty.tex: LP
92 Chapter 5. Uncertainty, Default, and Risk.



Important: When repayment is not certain, lenders demand a promised interest
rate that is higher than the expected interest rate by the default premium.


Promised Interest Rate
(5.19)
= Time Premium + Default Premium + Risk Premium .

The promised default premium is positive, but it is only paid when everything goes
well. The actually earned interest rate consists of the time premium, the realized
risk premium, and a (positive or negative) default realization.

Actual Interest Rate Earned
(5.20)
= Time Premium + Default Realization + Risk Premium .

The default realization could be more than negative enough to wipe out both the
time premium and the risk premium. But it is zero on average. Therefore,

Expected Interest Rate
(5.21)
= Time Premium + Expected Risk Premium .




The risk premium itself depends on such strange concepts as the correlation of loan default
Some real world
evidence. with the general economy and will be the subject of Part III of the book. However, we can preview
the relative importance of these components for you in the context of corporate bonds. (We
will look at risk categories of corporate bonds in more detail in the next chapter.) The highest-
quality bonds are called investment-grade. A typical such bond may promise about 6% per
annum, 150 to 200 basis points above the equivalent Treasury. The probability of default
would be small”less than 3% in total over a ten-year horizon (0.3% per annum). When an
investment-grade bond does default, it still returns about 75% of what it promised. For such
bonds, the risk premium would be small”a reasonable estimate would be that only about 10
to 20 basis points of the 200 basis point spread is the risk premium. The quoted interest rate
of 6% per annum therefore would re¬‚ect ¬rst the time premium, then the default premium,
and only then a small risk premium. (In fact, the liquidity premium would probably be more
important than the risk premium.) For low-quality corporate bonds, however, the risk premium
can be important. Ed Altman has been collecting corporate bond statistics since the 1970s. In
an average year, about 3.5% to 5.5% of low-grade corporate bonds defaulted. But in recessions,
the default rate shot up to 10% per year, and in booms it dropped to 1.5% per year. The average
value of a bond after default was only about 40 cents on the dollar, though it was as low 25
cents in recessions and as high as 50 cents in booms. Altman then computes that the most
risky corporate bonds promised a spread of about 5%/year above the 10-Year Treasury bond,
but ultimately delivered a spread of only about 2.2%/year. 280 points are therefore the default
premium. The remaining 220 basis points contain both the liquidity premium and the risk
premium”perhaps in roughly equal parts.
Solve Now!
Q 5.7 Return to the example in Table 5.2. Assume that the probability of receiving full payment
of $210 is only 95%, the probability of receiving $100 is 4%, and the probability of receiving
absolutely no payment is 1%. If the bond quotes a rate of return of 12%, what is the time premium,
the default premium and the risk premium?
¬le=uncertainty.tex: RP
93
Section 5·3. Uncertainty in Capital Budgeting, Debt, and Equity.

5·3. Uncertainty in Capital Budgeting, Debt, and Equity

We now turn to the problem of selecting projects under uncertainty. Your task is to compute
present values with imperfect knowledge about future outcomes. Your principal tool in this
task will be the payo¬ table (or state table), which assigns probabilities to the project value in
each possible future value-relevant scenario. For example, a ¬‚oppy disk factory may depend
on computer sales (say, low, medium, or high), whether ¬‚oppy disks have become obsolete
(yes or no), whether the economy is in a recession or expansion, and how much the oil price
(the major cost factor) will be. Creating the appropriate state table is the manager™s task”
judging how the business will perform depending on the state of these most relevant variables.
Clearly, it is not an easy task even to think of what the key variables are, to determine the
probabilities under which these variables will take on one or another value. Assessing how
your own project will respond to them is an even harder task”but it is an inevitable one. If
you want to understand the value of your project, you must understand what the project™s key
value drivers are and how the project will respond to these value drivers. Fortunately, for many
projects, it is usually not necessary to describe possible outcomes in the most minute detail”
just a dozen or so scenarios may be able to cover the most important information. Moreover,
these state tables will also allow you to explain what a loan (also called debt or leverage) and
levered ownership (also called levered equity) are, and how they di¬er.


5·3.A. Present Value With State-Contingent Payo¬ Tables

Almost all companies and projects are ¬nanced with both debt and levered equity. We already Most projects are
¬nanced with a mix of
know what debt is. Levered equity is simply what accrues to the business owner after the debt
debt and equity.
is paid o¬. (In this chapter, we shall not make a distinction between ¬nancial debt and other
obligations, e.g., tax obligations.) You already have an intuitive sense about this. If you own a
house with a mortgage, you really own the house only after you have made all debt payments.
If you have student loans, you yourself are the levered owner of your future income stream.
That is, you get to consume “your” residual income only after your liabilities (including your
non-¬nancial debt) are paid back. But what will the levered owner and the lender get if the
company™s projects fail, if the house collapses, or if your career takes a turn towards Rikers
Island? What is the appropriate compensation for the lender and the levered owner? The split
of net present value streams into loans (debt) and levered equity lies at the heart of ¬nance.
We will illustrate this split through the hypothetical purchase of a building for which the fu- The example of this
section: A building in
ture value is uncertain. This building is peculiar, though: it has a 20% chance that it will be
Tornado Alley can have
destroyed, say by a tornado, by next year. In this case, its value will only be the land”say, one of two possible
$20,000. Otherwise, with 80% probability, the building will be worth $100,000. Naturally, the future values.
$100,000 market value next year would itself be the result of many factors”it could include
any products that have been produced inside the building, real-estate value appreciation, as
well as a capitalized value that takes into account that a tornado might strike in subsequent
years.


Table 5.3. Building Payo¬ Table

Event Probability Value
Tornado 20% $20,000
Sunshine 80% $100,000
Expected Future Value $84,000
¬le=uncertainty.tex: LP
94 Chapter 5. Uncertainty, Default, and Risk.

The Expected Building Value
Table 5.3 shows the payo¬ table for full building ownership. The expected future building
To obtain the expected
future cash value of the value of $84,000 was computed as
building, multiply each
(possible) outcome by its
probability. E (Valuet=1 ) = ·
20% $20, 000

+ · = $84, 000
80% $100, 000
(5.22)
= Prob( Tornado ) · (Value if Tornadot=1 )

+ Prob( Sunshine ) · (Value if Sunshinet=1 ) .


Now, assume that the appropriate expected rate of return for a project of type “building” with
Then discount back the
expected cash value this type of riskiness and with one-year maturity is 10%. (This 10% discount rate is provided by
using the appropriate
demand and supply in the ¬nancial markets and known.) Your goal is to determine the present
cost of capital.
value”the appropriate correct price”for the building today.
There are two methods to arrive at the present value of the building”and they are almost
Under uncertainty, use
NPV on expected (rather identical to what we have done earlier. We only need to replace the known value with the
than actual, known) cash
expected value, and the known future rate of return with an expected rate of return. Now, the
¬‚ows, and use the
¬rst PV method is to compute the expected value of the building next period and to discount
appropriate expected
(rather than actual,
it at the cost of capital, here 10 percent,
known) rates of return.
The NPV principles
$84, 000
remain untouched. PVt=0 = ≈ $76, 363.64
1 + 10%
(5.23)
E (Valuet=1 )
= .
1 + E (rt=0,1 )




Table 5.4. Building Payo¬ Table, Augmented


Event Probability Value Discount Factor PV

1/(1+10%)
Tornado 20% $20,000 $18,181.82

1/(1+10%)
Sunshine 80% $100,000 $90,909.09




The second method is to compute the discounted state-contingent value of the building, and
Taking expectations and
discounting can be done then take expected values. To do this, augment Table 5.3. Table 5.4 shows that if the tornado
in any order.
strikes, the present value is $18,181.82. If the sun shines, the present value is $90,909.10.
Thus, the expected value of the building can also be computed

PVt=0 = ·
20% $18, 181.82

+ · ≈ $76, 363.64
80% $90, 909.09
(5.24)
= Prob( Tornado ) · (PV of Building if Tornado)

+ Prob( Sunshine ) · (PV of Building if Sunshine) .

Both methods lead to the same result: you can either ¬rst compute the expected value next year
(20% · $20, 000 + 80% · $100, 000 = $84, 000), and then discount this expected value of $84,000
to $76,363.34; or you can ¬rst discount all possible future outcomes ($20,000 to $18,181.82;
and $100,000 to $90,909.09), and then compute the expected value of the discounted values
(20% · $18, 181.82 + 80% · $90, 909.09 = $76, 363.34.)
¬le=uncertainty.tex: RP
95
Section 5·3. Uncertainty in Capital Budgeting, Debt, and Equity.



Important: Under uncertainty, in the NPV formula,

• known future cash ¬‚ows are replaced by expected discounted cash ¬‚ows, and
• known appropriate rates of return are replaced by appropriate expected
rates of return.

You can ¬rst do the discounting and then take expectations, or vice-versa.




The State-Dependent Rates of Return
What would the rates of return be in both states, and what would the overall expected rate of The state-contingent
rates of return can also
return be? If you have bought the building for $76,363.64, and no tornado strikes, your actual
be probability-weighted
rate of return (abbreviated rt=0,1 ) will be to arrive at the average
(expected) rate of
$100, 000 ’ $76, 363.64 return.
rt=0,1 = ≈ +30.95% . (5.25)
if Sunshine:
$76, 363.64

If the tornado does strike, your rate of return will be

$20, 000 ’ $76, 363.64
rt=0,1 = ≈ ’73.81% . (5.26)
if Tornado:
$76, 363.64

Therefore, your expected rate of return is

E (˜t=0,1 ) = ·
r (’73.81%)
20%

+ · = 10.00%
(+30.95%)
80%
(5.27)
E (˜t=0,1 ) = Prob( Tornado ) · (rt=0,1 if Tornado)
r

+ Prob( Sunshine ) · (rt=0,1 if Sunshine) .

The probability state-weighted rates of return add up to the expected overall rate of return.
This is as it should be: after all, we derived the proper price of the building today using a 10%
expected rate of return.
Solve Now!
Q 5.8 What changes have to be made to the NPV formula to handle an uncertain future?

Q 5.9 Under risk-neutrality, a factory can be worth $500,000 or $1,000,000 in two years, de-
pending on product demand, each with equal probability. The appropriate cost of capital is 6%
per year. What is the present value of the factory?

Q 5.10 A new product may be a dud (20% probability), an average seller (70% probability) or
dynamite (10% probability). If it is a dud, the payo¬ will be $20,000; if it is an average seller, the
payo¬ will be $40,000; if it is dynamite, the payo¬ will be $80,000. What is the expected payo¬
of the project?

Q 5.11 (Continued.) The appropriate expected rate of return for such payo¬s is 8%. What is the
PV of the payo¬?

Q 5.12 (Continued.) If the project is purchased for the appropriate present value, what will be
the rates of return in each of the three outcomes?

Q 5.13 (Continued.) Con¬rm the expected rate of return when computed from the individual
outcome speci¬c rates of return.
¬le=uncertainty.tex: LP
96 Chapter 5. Uncertainty, Default, and Risk.

5·3.B. Splitting Project Payo¬s into Debt and Equity

We now know how to compute the NPV of state-contingent payo¬s”our building paid o¬ dif-
State-contingent claims
have payoffs that ferently in the two states of nature. Thus, our building was a state-contingent claim”its payo¬
depend on future states
depended on the outcome. But it is just one of many. Another state-contingent claim might
of nature.
promise to pay $1 if the sun shines and $25 if a tornado strikes. Using payo¬ tables, we can
work out the value of any state-contingent claims”and in particular the value of the two most
important state-contingent claims, debt and equity.


The Loan
We now assume you want to ¬nance the building purchase of $76,363.64 with a mortgage of
Assume the building is
funded by a mortgagor $25,000. In e¬ect, the single project “building” is being turned into two di¬erent projects,
and a residual, levered
each of which can be owned by a di¬erent party. The ¬rst project is the project “Mortgage
building owner.
Lending.” The second project is the project “Residual Building Ownership,” i.e., ownership of
the building but bundled with the obligation to repay the mortgage. This “Residual Building
Ownership” investor will not receive a dime until after the debt has been satis¬ed. Such residual
ownership is called the levered equity, or just the equity, or even the stock in the building, in
order to avoid calling it “what™s-left-over-after-the-loans-have-been-paid-o¬.”
What sort of interest rate would the creditor demand? To answer this question, we need to
The ¬rst goal is to
determine the know what will happen if the building were to be condemned, because the mortgage value
appropriate promised
($25,000 today) will be larger than the value of the building if the tornado strikes ($20,000
interest rate on a
next year). We are assuming that the owner could walk away from it and the creditor could
“$25,000 value today”
mortgage loan on the
repossess the building, but not any of the borrower™s other assets. Such a mortgage loan is
building.
called a no-recourse loan. There is no recourse other than taking possession of the asset itself.
This arrangement is called limited liability. The building owner cannot lose more than the
money that he originally puts in. Limited liability is a mainstay of many ¬nancial securities: for
example, if you purchase stock in a company in the stock market, you cannot be held liable for
more than your investment, regardless of how badly the company performs.


Table 5.5. Payo¬ to Mortgage Creditor, Providing $25,000 Today


Prob
Event Value Discount Factor
1/(1+10%)
Tornado 20% $20,000
1/(1+10%)
Sunshine 80% Promised




Anecdote:
The framers of the United States Constitution had the English bankruptcy system in mind when they included
the power to enact “uniform laws on the subject of bankruptcies” in the Article I Powers of the legislative branch.
The ¬rst United States bankruptcy law, passed in 1800, virtually copied the existing English law. United States
bankruptcy laws thus have their conceptual origins in English bankruptcy law prior to 1800. On both sides of
the Atlantic, however, much has changed since then.
Early English law had a distinctly pro-creditor orientation, and was noteworthy for its harsh treatment of de-
faulting debtors. Imprisonment for debt was the order of the day, from the time of the Statute of Merchants
in 1285, until Dickens™ time in the mid-nineteenth century. The common law writs of capias authorized “body
execution,” i.e., seizure of the body of the debtor, to be held until payment of the debt.
English law was not unique in its lack of solicitude for debtors. History™s annals are replete with tales of
draconian treatment of debtors. Punishments in¬‚icted upon debtors included forfeiture of all property, re-
linquishment of the consortium of a spouse, imprisonment, and death. In Rome, creditors were apparently
authorized to carve up the body of the debtor, although scholars debate the extent to which the letter of that
law was actually enforced.
Direct Source: Charles Jordan Tabb, 1995, “The History of the Bankruptcy laws in the United States.”
www.bankruptcy¬nder.com/historyofbkinusa.html. (The original article contains many more juicy historical
tidbits.)
¬le=uncertainty.tex: RP
97
Section 5·3. Uncertainty in Capital Budgeting, Debt, and Equity.

To compute the PV for the project “Mortgage Lending,” we return to the problem of setting Start with the Payoff
Table, and write down
an appropriate interest rate, given credit risk (from Section 5·2). Start with the payo¬ table in
payoffs to project
Table 5.5. The creditor receives the property worth $20,000 if the tornado strikes, or the full “Mortgage Lending.”
promised amount (to be determined) if the sun shines. To break even, the creditor must solve
for the payo¬ to be received if the sun will shine in exchange for lending $25,000 today. This
is the “quoted” or “promised” payo¬.
$20, 000
= ·
$25, 000 20%
1 + 10%
Promise
+ ·
80%
1 + 10%
(5.28)
Loan Valuet=0 = Prob( Tornado ) · (Loan PV if Tornado)


+ Prob( Sunshine ) · (Loan PV if Sunshine) .

Solving, the solution is a promise of
(1 + 10%) · $25, 000 ’ 20% · $20, 000
Promise = = $29, 375
80%
(5.29)
[1 + E (r )] · Loan Value ’ Prob(Tornado) · Value if Tornado
=
Prob(Sunshine)

in repayment, paid by the borrower only if the sun will shine.
With this promised payo¬ of $29,375 (if the sun will shine), the lender™s rate of return will be The state-contingent
rates of return in the
the promised rate of return:
tornado (“default”)
state and in the sunshine
$29, 375 ’ $25, 000 state can be probability
rt=0,1 = = +17.50% , (5.30)
if Sunshine:
$25, 000 weighted to arrive at the
expected rate of return.
The lender would not provide the mortgage at any lower promised interest rate. If the tornado
strikes, the owner walks away, and the lender™s rate of return will be

$20, 000 ’ $25, 000
rt=0,1 = = ’20.00% . (5.31)
if Tornado:
$25, 000

Therefore, the lender™s expected rate of return is

E (˜t=0,1 ) = ·
r (’20.00%)
20%

+ · = 10.00%
(+17.50%)
80%
(5.32)
E (˜t=0,1 ) = Prob( Tornado ) · (rt=0,1 if Tornado)
r

+ Prob( Sunshine ) · (rt=0,1 if Sunshine) .

After all, in a risk-neutral environment, anyone investing for one year expects to earn an ex-
pected rate of return of 10%.


The Levered Equity
Our interest now turns towards proper compensation for you”the expected payo¬s and ex- Now compute the
payoffs of the 60%
pected rate of return for you, the residual building owner. We already know the building
post-mortgage (i.e.,
is worth $76,363.64 today. We also already know how the lender must be compensated: to levered) ownership of
contribute $25,000 to the building price today, you must promise to pay the lender $29,375 the building. The
method is exactly the
next year. Thus, as residual building owner, you need to pay $51,363.64”presumably from
same.
personal savings. If the tornado strikes, these savings will be lost and the lender will repos-
sess the building. However, if the sun shines, the building will be worth $100,000 minus the
promised $29,375, or $70,625. The owner™s payo¬ table in Table 5.6 allows you to determine
¬le=uncertainty.tex: LP
98 Chapter 5. Uncertainty, Default, and Risk.

that the expected future levered building ownership payo¬ is 20%·$0+80%·$70, 625 = $56, 500.
Therefore, the present value of levered building ownership is

$0 $70, 625
PVt=0 = 20% · + 80% ·
1 + 10% 1 + 10%
(5.33)
= Prob( Tornado ) · (PV if Tornado) + Prob( Sunshine ) · (PV if Sunshine)

= $51, 363.64 .




Table 5.6. Payo¬ To Levered Building (Equity) Owner

Prob
Event Value Discount Factor
1/(1+10%)
Tornado 20% $0.00
1/(1+10%)
Sunshine 80% $70,624.80




If the sun shines, the rate of return will be
Again, knowing the
state-contingent cash
$70, 624.80 ’ $51, 363.63
¬‚ows permits computing
rt=0,1 = = +37.50% . (5.34)
if Sunshine:
state-contingent rates of
$51, 363.63
return and the expected
rate of return.
If the tornado strikes, the rate of return will be
$0 ’ $51, 363.63
rt=0,1 = = ’100.00% . (5.35)
if Tornado:
$51, 363.63

The expected rate of return of levered equity ownership, i.e., the building with the bundled
mortgage obligation, is

E (˜t=0,1 ) = ·
r (’100.00%)
20%

+ · = 10.00%
(+37.50%)
80%
(5.36)
E (˜t=0,1 ) = Prob( Tornado ) · (rt=0,1 if Tornado)
r

+ Prob( Sunshine ) · (rt=0,1 if Sunshine) .




Re¬‚ections On The Example: Payo¬ Tables
Payo¬ tables are fundamental tools to think about projects and ¬nancial claims. You should
think about ¬nancial claims in terms of payo¬ tables.



Important: Whenever possible, in the presence of uncertainty, write down a
payo¬ table to describe the probabilities of each possible event (“state”) with its
state-contingent payo¬s.



Admittedly, this can sometimes be tedious, especially if there are many di¬erent possible states
(or even in¬nitely many states, as in a bell-shaped normally distributed project outcome”but
you can usually approximate even the most continuous and complex outcomes fairly well with
no more than ten discrete possible outcomes), but they always work!
¬le=uncertainty.tex: RP
99
Section 5·3. Uncertainty in Capital Budgeting, Debt, and Equity.



Table 5.7. Payo¬ Table and Overall Values and Returns


Prob
Event Building Value Mortgage Value Levered Ownership
Tornado 20% $20,000 $20,000 $0
Sunshine 80% $100,000 $29,375 $70,625
Expected Valuet=1 $84,000 $27,500 $56,500
PVt=0 $76,364 $25,000 $51,364
E(rt=0,1 ) 10% 10% 10%




Table 5.7 shows how elegant such a table can be. It can describe everything we need in a There are three possible
investment
very concise manner: the state-contingent payo¬s, expected payo¬s, net present value, and
opportunities here. The
expected rates of return for your building scenario. Because owning the mortgage and the bank is just another
levered equity is the same as owning the full building, the last two columns must add up to the investor, with particular
payoff patterns.
values in the “building value” column. You could decide to be any kind of investor: a creditor
(bank) who is loaning money in exchange for promised payment; a levered building owner who
is taking a “piece left over after a loan”; or an unlevered building owner who is investing money
into an unlevered project. You might take the whole piece (that is, 100% of the claim”all three
investments are just claims) or you might just invest, say $5, at the appropriate fair rates of
return that are due to investors in our perfect world, where everything can be purchased at or
sold at a fair price. (Further remaining funds can be raised elsewhere.)


Re¬‚ections On The Example and Debt and Equity Risk
We have not covered risk yet, because we did not need to. In a risk-neutral world, all that
matters is the expected rate of return, not how uncertain you are about what you will receive.
Of course, we can assess risk even in a risk-neutral world, even if risk were to earn no extra
compensation (a risk premium).
So, which investment is most risky: full ownership, loan ownership, or levered ownership? Leveraging (mortgaging)
a project splits it into a
Figure 5.2 plots the histograms of the rates of return to each investment type. As the visual
safer loan and a riskier
shows, the loan is least risky, followed by the full ownership, followed by the levered ownership. levered ownership,
Your intuition should tell you that, by taking the mortgage, the medium-risky project “building” although everyone
expects to receive 10%
has been split into a more risky project “levered building” and a less risky project “mortgage.”
on average.
The combined “full building ownership” project therefore has an average risk.
It should not come as a surprise to learn that all investment projects expect to earn a 10% If everyone is
risk-neutral, everyone
rate of return. After all, 10% is the time-premium for investing money. Recall from Page 92
should expect to earn
that the expected rate of return (the cost of capital) consists only of a time-premium and a 10%.
risk premium. (The default premium is a component only of promised interest rates, not of
expected interest rates; see Section 5·2). By assuming that investors are risk-neutral, we have
assumed that the risk premium is zero. Investors are willing to take any investment that o¬ers
an expected rate of return of 10%, regardless of risk.
Although our example has been a little sterile, because we assumed away risk preferences, it is Unrealistic, maybe! But
ultimately, maybe not.
nevertheless very useful. Almost all projects in the real world are ¬nanced with loans extended
by one party and levered ownership held by another party. Understanding debt and equity is
as important to corporations as it is to building owners. After all, stocks in corporations are
basically levered ownership claims that provide money only after the corporation has paid back
its loans. The building example has given you the skills to compute state-contingent, promised,
and expected payo¬s, and state-contingent, promised, and expected rates of returns”the nec-
essary tools to work with debt, equity, or any other state-contingent claim. And really, all that
will happen later when we introduce risk aversion is that we will add a couple of extra basis
¬le=uncertainty.tex: LP
100 Chapter 5. Uncertainty, Default, and Risk.




Figure 5.2. Probability Histograms of Project Returns




1.0
0.8
Histogram of Rate of Re-




0.6
Probability
turn for Project of Type




0.4
“Full Ownership”




0.2
0.0
’100 ’50 0 50 100

Rate of Return, in %




1.0
0.8
Histogram of Rate of Re-




0.6
Probability
turn for Project of Type




0.4
“Levered Ownership”



0.2
(Most Risky)


0.0
’100 ’50 0 50 100

Rate of Return, in %
1.0
0.8




Histogram of Rate of Re-
0.6
Probability




turn for Project of Type
0.4




“Loan Ownership”
0.2




(Least Risky)
0.0




’100 ’50 0 50 100

Rate of Return, in %




20% probability 80% probability Expected

Tornado Sunshine Average Variability
$20, 000 $29, 375 $27, 500
Loan (Ownership) ’1= ’1= ’1=
$25, 000 $25, 000 $25, 000

’20.00% +17.50% ≈ ±20%
10.00%
$0 $70, 625 $56, 500
Levered Ownership ’1= ’1= ’1=
$51, 364 $51, 364 $51, 364

’100.00% +37.50% ≈ ±60%
10.00%
$20, 000 $100, 000 $84, 000
Full Ownership ’1= ’1= ’1=
$76, 364 $76, 364 $76, 364

’73.81% +30.95% ≈ ±46%
10.00%

Consider the variability number here to be just an intuitive measure”it is the aforementioned “standard deviation”
that will be explained in great detail in Chapter 13. Here, it is computed as

20% · (’20.00% ’ 10%)2 + 80% · (17.50% ’ 10%)2
Sdv = = 19.94%
Loan Ownership

20% · (’100.00% ’ 10%)2 + 80% · (37.50% ’ 10%)2 = 59.04%
Levered Ownership Sdv =
(5.37)
20% · (’73.81% ’ 10%)2 + 80% · (30.95% ’ 10%)2
Sdv = = 46.07%
Full Ownership

Prob T · [VT ’ E (V )]2 + Prob S · [VS ’ E (V )]2
Sdv = .
¬le=uncertainty.tex: RP
101
Section 5·3. Uncertainty in Capital Budgeting, Debt, and Equity.

points of required compensation”more to equity (the riskiest claim) than to the project (the
medium-risk claim) than to debt (the safest claim).
Solve Now!
Q 5.14 In the example, the building was worth $76,364, the mortgage was worth $25,000, and
the equity was worth $51,364. The mortgage thus ¬nanced 32.7% of the cost of the building, and
the equity ¬nanced $67.3%. Is the arrangement identical to one in which two partners purchase
the building together”one puts in $25,000 and owns 32.7%, and the other puts in $51,364 and
owns 67.3%?


Q 5.15 Buildings are frequently ¬nanced with a mortgage that pays 80% of the price, not just
32.7% ($25,000 of $76,364). Produce a table similar to Table 5.7 in this case.


Q 5.16 Repeat the example if the loan does not provide $25,000, but promises to pay o¬ $25,000.
How much money do you get for this promise? What is the promised rate of return. How does the
riskiness of the project “full building ownership” compare to the riskiness of the project “levered
building ownership”?


Q 5.17 Repeat the example if the loan promises to pay o¬ $20,000. Such a loan is risk-free. How
does the riskiness of the project “full building ownership” compare to the riskiness of the project
“levered building ownership”?


Q 5.18 Under risk-neutrality, a factory can be worth $500,000 or $1,000,000 in two years, de-
pending on product demand, each with equal probability. The appropriate cost of capital is 6%
per year. The factory can be ¬nanced with proceeds of $500,000 from loans today. What are
the promised and expected cash ¬‚ows and rates of return for the factory (without loan), for the
loan, and for a hypothetical factory owner who has to ¬rst repay the loan?


Q 5.19 Advanced: For illustration, we assumed that the sample building was not lived in. It
consisted purely of capital amounts. But, in the real world, part of the return earned by a
building owner is rent. Now assume that rent of $11,000 is paid strictly at year-end, and that
both the state of nature (tornado or sun) and the mortgage loan payment happens only after the
rent has been safely collected. The new building has a resale value of $120,000 if the sun shines,
and a resale value of $20,000 if the tornado strikes.

(a) What is the value of the building today?

(b) What is the promised interest rate for a lender providing $25,000 in capital today?

(c) What is the value of residual ownership today?

(d) Conceptual Question: What is the value of the building if the owner chooses to live in the
building?
¬le=uncertainty.tex: LP
102 Chapter 5. Uncertainty, Default, and Risk.

More Than Two Possible Outcomes
How does this example generalize to multiple possible outcomes? For example, assume that the
Multiple outcomes will
cause multiple building could be worth $20,000, $40,000, $60,000, $80,000, or $100,000 with equal probability,
breakpoints.
and the appropriate expected interest rate were 10%”so the building has an PV of $60, 000/(1+
10%) ≈ $54, 545.45. If a loan promised $20,000, how much would you expect to receive? But,
of course, $20,000!
E Payo¬(Loan Promise =$20,000) = $20, 000 .
(5.38)
Payo¬ of Loan
E = .
Loan
if $0 ¤ Loan ¤ $20, 000
If a loan promised $20,001, how much would you expect to receive? $20,000 for sure, plus the
extra “marginal” $1 with 80% probability. In fact, you would expect only 80 cents for each dollar
promised between $20,000 and $40,000. So, if a loan promised $40,000, you would expect to
receive

E = $20, 000 + 80% · ($40, 000 ’ $20, 000)
Payo¬( Loan Promise = $40,000 )

(5.39)
= $36, 000
Payo¬ of Loan
E = $20, 000 + 80% · (Loan ’ $20, 000) .
if $20, 000 ¤ Loan ¤ $40, 000

If a loan promised you $40,001, how much would you expect to receive? You would get $20,000
for sure, plus another $20,000 with 80% probability (which is an expected $16,000), plus the
marginal $1 with only 60% probability. Thus,

E Payo¬(Loan Promise = $40,001) = + 80% · ($40, 000 ’ $20, 000)
$20, 000

+ 60% · $1

(5.40)
= $36, 000.60
Payo¬ of Loan
E = + 80% · $20, 000
$20, 000
if $40, 000 ¤ Loan ¤ $60, 000
+ 60% · (Loan ’ $40, 000) .

And so on. Figure 5.3 plots these expected payo¬s as a function of the promised payo¬s. With
this ¬gure, mortgage valuation becomes easy. For example, how much would the loan have to
promise to provide $35,000 today? The expected payo¬ would have to be (1+10%)·$35, 000 =
$38, 500. Figure 5.3 shows that an expected payo¬ of $38,500 corresponds to around $44,000
in promise. (The exact number can be worked out to be $44,167.) Of course, we cannot borrow
more than $54,545.45, the project™s PV. So, we can forget about the idea of obtaining a $55,000
mortgage.
Solve Now!
Q 5.20 What is the expected payo¬ if the promised payo¬ is $45,000?


Q 5.21 What is the promised payo¬ if the expected payo¬ is $45,000?


Q 5.22 Assume that the probabilities are not equal: $20,000 with probability 12.5%, $40,000
with probability 37.5%, $60,000 with probability 37.5%, and $80,000 with probability 12.5%.

(a) Draw a graph equivalent to Figure 5.3.

(b) If the promised payo¬ of a loan is $50,000, what is the expected payo¬?

(c) If the prevailing interest rate is 5% before loan payo¬, then how much repayment does a
loan providing $25,000 today have to promise? What is the interest rate?

You do not need to calculate these values, if you can read them o¬ your graph.
¬le=uncertainty.tex: RP
103
Section 5·3. Uncertainty in Capital Budgeting, Debt, and Equity.



Figure 5.3. Promised vs. Expected Payo¬s




60
50
40
Expect

30
20
10
0




0 20 40 60 80 100

Promise




Q 5.23 A new product may be a dud (20% probability), an average seller (70% probability) or
dynamite (10% probability). If it is a dud, the payo¬ will be $20,000; if it is an average seller, the
payo¬ will be $40,000; if it is dynamite, the payo¬ will be $80,000. The appropriate expected
rate of return is 6% per year. If a loan promises to pay o¬ $40,000, what are the promised and
expected rates of return?


Q 5.24 Advanced: What is the formula equivalent to (5.40) for promised payo¬s between $60,000
and $80,000?


Q 5.25 Advanced: Can you work out the exact $44,167 promise for the $35,000 (today!) loan?
¬le=uncertainty.tex: LP
104 Chapter 5. Uncertainty, Default, and Risk.

5·4. Robustness: How Bad are Your Mistakes?

Although it would be better to get everything perfect, it is often impossible to come up with
How bad are mistakes?
perfect cash ¬‚ow forecasts and appropriate interest rate estimates. Everyone makes errors.
So, how bad are mistakes? How robust is the NPV formula? Is it worse to commit an error in
estimating cash ¬‚ows or in estimating the cost of capital? To answer these questions, we will
do a simple form of scenario analysis”we will consider just a very simple project, and see
how changes in estimates matter to the ultimate value. Doing good scenario analysis is also
good practice for any managers”so that they can see how sensitive their estimated value is to
reasonable alternative possible outcomes. Therefore this method is also called a sensitivity
analysis. Doing such analysis becomes even more important when we consider “real options”
in our next chapter.


5·4.A. Short-Term Projects

Assume that your project will pay o¬ $200 next year, and the proper interest rate for such
The benchmark case: A
short-term project, projects is 8%. Thus, the correct project present value is
correctly valued.
$200
(5.44)
PVcorrect = ≈ $185.19 .
1 + 8%


If you make a 10% error in your cash ¬‚ow, e.g., mistakenly believing it to return $220, you will
Committing an error in
cash ¬‚ow estimation. compute the present value to be

$220
(5.45)
PVCF error = ≈ $203.70 .
1 + 8%
The di¬erence between $203.70 and $185.19 is a 10% error in your present value.
In contrast, if you make a 10% error in your cost of capital (interest rate), mistakenly believing
Committing an error in
interest rate estimation. it to require a cost of capital (expected interest rate) of 8.8% rather than 8%, you will compute
the present value to be
$200
(5.46)
= ≈ $183.82 .
PVr error
1 + 8.8%
The di¬erence between $183.82 and $185.19 is less than a $2 or 1% error.


Important: For short-term projects, errors in estimating correct interest rates
are less problematic in computing NPV than are errors in estimating future cash
¬‚ows.




5·4.B. Long-Term Projects

Now take the same example, but assume the cash ¬‚ow will occur in 30 years. The correct
A long-term project,
correctly valued and present value is now
incorrectly valued.
$200
PVcorrect = ≈ $19.88 (5.47)
.
(1 + 8%)30
The 10% “cash ¬‚ow error” present value is

$220
PVCF error = ≈ $21.86 (5.48)
,
(1 + 8%)30

and the 10% “interest rate error” present value is

$200
= ≈ $15.93 (5.49)
.
PVr error
(1 + 8.8%)30
¬le=uncertainty.tex: RP
105
Section 5·4. Robustness: How Bad are Your Mistakes?.

This calculation shows that cash ¬‚ow estimation errors and interest rate estimation errors are Both cash ¬‚ow and cost
of capital errors are now
now both important. So, for longer-term projects, estimating the correct interest rate becomes
important.
relatively more important. However, in fairness, estimating cash ¬‚ows thirty years into the
future is just about as di¬cult as reading a crystal ball. (In contrast, the uncertainty about the
long-term cost of capital tends not explode as quickly as your uncertainty about cash ¬‚ows.)
Of course, as di¬cult as it may be, we have no alternative. We must simply try to do our best
at forecasting.



Important: For long-term projects, errors in estimating correct interest rates
and errors in estimating future cash ¬‚ows are both problematic in computing NPV.




5·4.C. Two Wrongs Do Not Make One Right

Please do not think that you can arbitrarily adjust an expected cash ¬‚ow to paint over an issue Do not think two errors
cancel.
with your discount rate estimation, or vice-versa. For example, let us presume that you consider
an investment project that is a bond issued by someone else. You know the bond™s promised
payo¬s. You know these are higher than the expected cash ¬‚ows. Maybe you can simply use
the average promised discount rate on other risky bonds to discount the bond™s promised cash
¬‚ows? After all, the latter also re¬‚ects default risk. The two default issues might cancel one
another, and you might end up with the correct number. Or they might not cancel and you end
up with a non-sense number!
Let™s say the appropriate expected rate of return is 10%. A suggested bond investment may An example: do not
think you can just work
promise $16,000 for a $100,000 investment, but have a default risk on the interest of 50% (the
with promised rates.
principal is insured). Your benchmark promised opportunity cost of capital may rely on risky
bonds that have default premia of 2%. Your project NPV is neither ’$100, 000 + $116, 000/(1 +
12%) ≈ $3, 571 nor ’$100, 000 + $100, 000/(1 + 10%) + $16, 000/(1 + 12%) ≈ $5, 195. Instead,
you must work with expected values

$100, 000 $8, 000
(5.50)
PV = ’$100, 000 + + ≈ ’$1, 828 .
1 + 10% 1 + 10%
This bond would be a bad investment.
Solve Now!
Q 5.26 What is the relative importance of cash ¬‚ow and interest rate errors for a 10-year project?


Q 5.27 What is the relative importance of cash ¬‚ow and interest rate errors for a 100-year
project?
¬le=uncertainty.tex: LP
106 Chapter 5. Uncertainty, Default, and Risk.

5·5. Summary

The chapter covered the following major points:

• The possibility of future default causes promised interest rates to be higher than expected
interest rates. Default risk is also often called credit risk.

• Quoted interest rates are almost always promised interest rates, and are higher than
expected interest rates.

• Most of the di¬erence between promised and expected interest rates is due to default.
Extra compensation for bearing more risk”the risk premium”is typically much smaller
than the default premium.

• The key tool for thinking about uncertainty is the payo¬ table. Each row represents one
possible state outcome, which contains the probability that the state will come about, the
total project value that can be distributed, and the allocation of this total project value
to di¬erent state-contingent claims. The state-contingent claims “carve up” the possible
project payo¬s.

• Most real-world projects are ¬nanced with the two most common state-contingent claims”
debt and equity. The conceptual basis of debt and equity is ¬rmly grounded in payo¬
tables. Debt ¬nancing is the safer investment. Equity ¬nancing is the riskier investment.

• If debt promises to pay more than the project can deliver in the worst state of nature, then
the debt is risky and requires a promised interest rate in excess of its expected interest
rate.

• NPV is robust to uncertainty about the expected interest rate (the discount rate) for short-
term projects. However, NPV is not robust with respect to either expected cash ¬‚ows or
discount rates for long-run projects.
¬le=uncertainty.tex: RP
107
Section 5·5. Summary.

Solutions and Exercises




1. No! It is presumed to be known”at least for a die throw. The following is almost philosophy and beyond
what you are supposed to know or answer here: It might, however, be that the expected value of an investment
is not really known. In this case, it, too, could be a random variable in one sense”although you are presumed
to be able to form an expectation (opinion) over anything, so in this sense, it would not be a random variable,
either.
2. Yes and no. If you do not know the exact bet, you may not know the expected value.
3. If the random variable is the number of dots on the die, then the expected outcome is 3.5. The realization
was 6.
4. The expected value of the stock is $52. Therefore, purchasing the stock at $50 is not a fair bet, but a good
bet.


5. Only for government bonds. Most other bonds have some kind of default risk.
6.
(a) The expected payo¬ is now 95%·$210 + 1%·$100 + 4%·$0 = $200.50. Therefore, the expected rate of
return is $200.50/$200 = 0.25%.
(b) You require an expected payo¬ of $210. Therefore, you must solve for a promised payment 95%·P +
1%·$100 + 4%·$0 = $210 ’ P = $209/0.95 = $220. On a loan of $200, this is a 10% promised interest
rate.
7. The expected payo¬ is $203.50, the promised payo¬ is $210, and the stated price is $210/(1+12%)=$187.50.
The expected rate of return is $203.50/$187.50 = 8.5%. Given that the time premium, the Treasury rate is 5%,
the risk premium is 3.5%. The remaining 12%-8.5%=3.5% is the default premium.


8. The actual cash ¬‚ow is replaced by the expected cash ¬‚ow, and the actual rate of return is replaced by the
expected rate of return.
9. $750, 000/(1 + 6%)2 ≈ $667, 497.33.
10. E (P ) = 20% · $20, 000 + 70% · $40, 000 + 10% · $80, 000 = $40, 000.
11. $37, 037.
12. $20, 000/$37, 037 ’ 1 = ’46%, $40, 000/$37, 037 ’ 1 = +8%, $80, 0000/$37037 ’ 1 = +116%.
13. 20%·(’46%) + 70%·(+8%) + 10%·(+116%) = 8%.
14. No! Partners would share payo¬s proportionally, not according to “debt comes ¬rst.” For example, in the
tornado state, the 32.7% partner would receive only $6,547.50, not the entire $20,000 that the debt owner
receives.
15. The mortgage would ¬nance $61,090.91 today.
Prob
Event Building Value Mortgage Value Levered Ownership
Tornado 20% $20,000 $20,000 $0
Sunshine 80% $100,000 $79,000 $21,000
Expected Valuet=1 $84,000 $67,200 $16,800
PVt=0 $76,364 $61,091 $15,273
E (rt=0,1 ) 10% 10% 10%
16. In the tornado state, the creditor gets all ($20,000). In the sunshine state, the creditor receives the promise
of $25, 000. Therefore, the creditor™s expected payo¬ is 20% · $20, 000 + 80% · $25, 000 = $24, 000. To o¬er
an expected rate of return of 10%, you can get $24, 000/1.1 = $21, 818 from the creditor today. The promised
rate of return is therefore $25, 000/$21, 818 ’ 1 = 14.58%.
17. The loan pays o¬ $20,000 for certain. The levered ownership pays either $0 or $80,000, and costs $64, 000/(1+
10%) = $58, 182. Therefore, the rate of return is either ’100% or +37.5%. We have already worked out full
ownership. It pays either $20,000 or $100,000, costs $76,364, and o¬ers either ’73.81% or +30.95%. By
inspection, the levered equity project is riskier. In e¬ect, building ownership has become riskier, because the
owner has chosen to sell o¬ the risk-free component, and retain only the risky component.
18. Factory: The expected factory value is $750,000. Its price would be $750, 000/1.062 = $667, 497. The
promised rate of return is therefore $1, 000, 000/$667, 497 ’ 1 ≈ 49.8%. Loan: The discounted (today™s) loan
price is $750, 000/1.062 = $667, 497.33. The promised value is $1,000,000. The loan must have an expected
payo¬ of 1.062 · $500, 000 = $561, 800 (6% expected rate of return, two years). Because the loan can pay
$500,000 with probability 1/2, it must pay $623,600 with probability 1/2 to reach $561,800 as an average.
Therefore, the promised loan rate of return is $623, 600/$500, 000 ’ 1 = 24.72% over two years (11.68%
per annum). Equity: The levered equity must therefore pay for/be worth $667, 497.33 ’ $500, 000.00 =
$167, 497.33 (alternatively, levered equity will receive $1, 000, 000 ’ $623, 600 = $376, 400 with probability
1/2 and $0 with probability 1/2), for an expected payo¬ of $188,200. (The expected two-year holding rate
of return is $188, 200/$167, 497 ’ 1 = 12.36% [6% per annum, expected].) The promised rate of return is
($1, 000, 000 ’ $623, 600)/$167, 497.33 ’ 1 = 124.72% (50% promised per annum).
¬le=uncertainty.tex: LP
108 Chapter 5. Uncertainty, Default, and Risk.

19.
(a) In the sun state, the value is $120,000+$11,000= $131,000. In the tornado state, the value is $11,000+$20,000=
$31,000. Therefore, the expected building value is $111,000. The discounted building value today is
$100,909.09.
(b) Still the same as in the text: the lender™s $25,000 loan can still only get $20,000, so it is a promise for
$29,375. So the quoted interest rate is still 17.50%.
(c) $100, 909.09 ’ $25, 000 = $75, 909.09.
(d) Still $100,909.09, assuming that the owner values living in the building as much as a tenant would.
Owner-consumed rent is the equivalent of corporate dividends paid out to levered equity. Note: you can repeat
this example assuming that the rent is an annuity of $1,000 each month, and tornadoes strike mid-year.
20. From the graph, it is around $40,000. The correct value can be obtained by plugging into Formula (5.40):
$39,000.
21. From the graph, it is around $55,000. The correct value can be obtained by setting Formula (5.40) equal to
$55,000 and solving for “Loan.” The answer is indeed $55,000.
22.
60
50
40
Expect

30
20
10
0




0 20 40 60 80 100

Promise
(a)
(b) The exact expected payo¬ is 1/8 · $20, 000 + 3/8 · $40, 000 + 1/2 · $50, 000 = $42, 500. The 1/2 is the
probability that you will receive the $50,000 that you have been promised, which occurs if the project
ends up worth at least as much as your promised $50,000. This means that it is the total probability
that it will be worth $60,000 or $80,000.
(c) The loan must expect to pay o¬ (1 + 5%) · $25, 000 = $26, 250. Therefore, solve 1/8 · $20, 000 + 7/8 · x =
$26, 250, so the exact promised payo¬ must be x = $27, 142.90.

23. With 20% probability, the loan will pay o¬ $20,000; with 80% probability, the loan will pay o¬ the full promised
$40,000. Therefore, the loan™s expected payo¬ is 20%·$20, 000 + 80%·$40, 000 = $36, 000. The loan™s price
is $36, 000/(1 + 6%) = $33, 962. Therefore, the promised rate of return is $40, 000/$33, 962 ’ 1 ≈ 17.8%. The
expected rate of return was given: 6%.
24.
Payo¬ of Loan
E = $20, 000
if $60, 000 ¤ Loan ¤ $80, 000
+ 80% · $20, 000
(5.51)
+ 60% · $20, 000

+ 40% · (Loan ’ $60, 000) .

25. The loan must yield an expected value of $38,500. Set formula (5.40) equal to $38,500 and solve for “Loan.”
The answer is indeed $44,166.67.
¬le=uncertainty.tex: RP
109
Section 5·5. Summary.

26. Consider a project that earns $100 in 10 years, and where the correct interest rate is 10%.
• The correct PV is $100/(1 + 10%)10 = $38.55.
• If the cash ¬‚ow is incorrectly estimated to be 10% higher, the incorrect PV is $110/(1 + 10%)10 = $42.41.
• If the interest rate is incorrectly estimate to be 10% lower, the incorrect PV is $100/(1 + 9%)10 = $42.24.
So, the misvaluation e¬ects are reasonably similar at 10% interest rates. Naturally, percent valuation mistakes
in interest rates are higher if the interest rate is higher; and lower if the interest rate is lower.
27. Although this, too, depends on the interest rate, interest rate errors almost surely matter for any reasonable
interest rates now.



(All answers should be treated as suspect. They have only been sketched, and not been checked.)
¬le=uncertainty.tex: LP

<<

. 6
( 39)



>>